Ultrahigh‐Resolution Optical Fiber Thermometer Based on Microcavity Opto‐Mechanical Oscillation

High‐resolution temperature measurement is nerve‐wracking obstruction for precise characterization of many physical, chemical, and biological processes. To solve this problem, a novel microcavity–optomechanical–oscillation‐based thermometer is proposed. The microcavity serving as a link parametrically couples the mechanical resonator and optical resonator in the same structure and provides a natural and highly sensitive temperature transduction mechanism and ultrahigh‐resolution optical demodulation. The mathematical model of geometrical parameters, mechanics, and material properties for temperature response mechanism is established and verified experimentally. The proposed thermometer has a thermal sensitivity of 11 300 Hz °C−1 and an ultrahigh‐temperature resolution of 1 × 10−4 °C, to the best of one's knowledge, which is the highest temperature resolution with a silica cavity.


Introduction
Temperature is a fundamental physical property for characterizing many reactions or states in physical, biological, and chemical processes. High-resolution temperature-sensing provides a powerful boost in many fields. For instance, precise and continuous monitoring of temperature changes in the ocean is critical for understanding the thermal phenomenon and providing essential observations to improve climate models, such as dynamic ocean circulation [1] or suppression of global warming. [2] Another example is that the friction dissipates heat during an earthquake, the fault temperature after an earthquake provides insight into the level of friction and controls earthquake dynamics. [3,4] The resolution of temperature sensing in ocean and geological changes needs to be order of at least %10 À3°C . [2,3] In addition, semiconductor industry, [5,6] biology, medicine, [7][8][9][10] and energy harvesting [11,12] also require ultrahightemperature resolving ability.
Various thermal-sensing methods based on optical fiber sensors are developed due to its electromagnetic immunity and electrical passive, such as fiber Bragg gratings and fiber Fabry-Perot interferometers. However, the requirement of high resolution is not well resolved. Optical resonant cavities [13] provide a good platform solution. Especially, the whispering gallery mode (WGM) resonators [14,15] can provide high Q value and strong light-matter interaction, consequently improving resolution. The temperature variation induces a change in the cavity optical path, which causes the sharp resonant dips to move. Temperature variation information is obtained by tracking the resonant wavelengths shift or monitoring the patterns of multiple modes. [16] Moreover, the use of multiple modes may realize simultaneous measurement of multiple parameters, such as a pair of cavity modes are used to decouple the refractive index and temperature information of the analyte during the phase-transition process. [17] Starting from solid silica microcavities, [18] organic materials microcavities, [19][20][21] all-liquid microcavities, [22,23] and hollow liquid core microcavities [24,25] have been developed for temperature sensing. The advantages of organic material and liquid microcavities are their large thermal expansion coefficient and thermo-optical coefficient, so they have higher temperature sensitivity than silica microcavities. However, organic material microcavities [26] show a slower response due to poor thermal conductivity. In addition, the consistency between heating and cooling cycle is worse due to large hysteresis essence of organic material. Liquid microcavities are difficult to manipulate and store, which prevent it from stable and long-term monitoring. Its large cavity optical loss also deteriorates the Q value and sensing resolution.
Recently, the cavity optomechanical systems [27,28] have attracted increasing attention due to their enhanced interactions between light resonator mode and mechanical resonator mode in a single microcavity. WGM-resonator-based optomechanics [29][30][31] have been used for many fundamental experiments including mesoscopic quantum mechanics studying, [32] intrinsic cavity cooling, [33,34] and chaotic quivering. [35] The mechanical resonance can enhance response and optical resonance can enhance readout sensitivity with less detrimental noise, [36] therefore cavity optomechanical systems have a nature essence of high-resolution transduction mechanism for bioparticles, [37] viscous liquid DOI: 10.1002/adpr.202200052 High-resolution temperature measurement is nerve-wracking obstruction for precise characterization of many physical, chemical, and biological processes. To solve this problem, a novel microcavity-optomechanical-oscillation-based thermometer is proposed. The microcavity serving as a link parametrically couples the mechanical resonator and optical resonator in the same structure and provides a natural and highly sensitive temperature transduction mechanism and ultrahigh-resolution optical demodulation. The mathematical model of geometrical parameters, mechanics, and material properties for temperature response mechanism is established and verified experimentally. The proposed thermometer has a thermal sensitivity of 11 300 Hz°C À1 and an ultrahightemperature resolution of 1 Â 10 À4°C , to the best of one's knowledge, which is the highest temperature resolution with a silica cavity.
analysis, [38,39] mass, and displacement. [40,41] However, there are few explorations on temperature sensing using optomechanical oscillation. In fact, the previous researches also need to consider effect of tiny temperature fluctuation.
In this paper, we propose and experimentally demonstrate an ultrahigh-resolution optical fiber thermometer based on radiation-pressure-driven optomechanical oscillation on a hollow silica microbubble. The cavity serving as a link parametrically couples the mechanical resonator and optical resonator in the same structure, and provides a natural and highly sensitive temperature transduction mechanism and ultrahigh-resolution optical demodulation. A stable optomechanical oscillation with oscillation frequency of %10 MHz is generated on a self-made silica microbubble with outer diameter of %100 μm. The influences of geometrical parameters, mechanics, and material properties on temperature response mechanism are investigated theoretically and experimentally. The optical fiber thermometer is interrogated with electrical domain frequency shift of opto-mechanical oscillation, which avoids expensive ultrahighresolution optical spectral demodulation. An ultrahightemperature resolution of 1 Â 10 À4°C is achieved, which is the highest-temperature resolution with silica cavity, to our best knowledge.

Principle of Temperature Sensing Based on Microcavity Optomechanical Oscillation
The temperature sensor comprises an ultrathin-wall microbubble formed on a silica microcapillary and fixed on a copper sheet by UV glue, as shown in Figure 1a. A microbubble works as a mechanical resonator and an optical resonator at the same time. The tiny temperature change will affect the mechanical-mode frequency through geometrical parameters, mechanics, and material properties of microbubble. The reading out mechanism of optomechanical oscillation is shown in Figure 1b,c. The microbubble supports an initial WGM resonance mode with optical frequency ω 0 . A continuous-wave optical input pump with optical frequency ω (higher than ω 0 ) is coupled into the microbubble through a tapered fiber and generates radiation pressure, which expands the cavity structure and consequently decreases WGM optical frequency ω 0 , that is, spectral line with black color shifts to red one in Figure 1c. The WGM optical frequency ω 0 shift to ω 0 0 will cause larger optical loss and reduce circulating optical power in the microcavity, which in turn reduce radiation pressure and make microbubble restore the mechanical deformation. When positive feedback produced by the circulating optical power is large enough to overcome the mechanical loss, a periodic deformation motion or breathing mechanical oscillation of microbubble at mechanical eigenmode frequency f will be excited. The modulated output light shows the same amplitude modulation frequency f (Figure 1c inset). Thus, the temperature is reflected in output light-modulated frequency, which can be easily obtained from a low-cost electrical domain spectrometer or Fourier transfer of time-domain data acquisition. Figure 1d illustrates the opto-mechanical sensing path.
Establishing a mechanical resonator analytic model will help the sensor design. However, it is hard to get an analytic mechanical solution for a complicated structure. Considering the tiny fiber taper being used to excite the microbubble vertically, outer diameter of microbubble is far larger than that of fiber taper and the axial dimension (%cm) of sensor is far larger than its transverse dimension (%100 μm), and it will be a reasonable simplified model by approximating the sensor with cylindrical shell model. The breathing mode mechanical resonant frequency of the sensor can be described as [42][43][44] where R is the outer radius, H is the wall thickness, and L is the distance between two UV glue points. E is the Young's modulus and ρ is the mass density. n is the number of circumferential waves. D is bending stiffness factor, given by are the axial and the circumferential resultant stresses, frequency is designated as initial frequency f 0 . When the temperature changes, four dominating factors, that is, axial force variation ΔF (through N xF ), internal pressure variation ΔP in (through N xP and N φP ), microbubble radius variation ΔR, and Young's modulus variation ΔE (silica Young's modulus temperature coefficient ΔE si is 183 ppm°C À1 ), [45] will change mechanical resonant frequency of the sensor (see Figure 1a) and therefore its temperature response. The temperature sensitivity is given by There is large difference between the thermal expansion coefficient of copper sheet and that of silica, so we enhance the temperature response of the sensor greatly by fixing the both ends of microcapillary on copper sheet base to produce large thermal-induced stress. The axial force ΔF will generate axial stress on the microbubble.
where A si is the microbubble cross-sectional area.
Although the ends of microcapillary are not sealed, the flow resistance due to its small diameter makes the air unable to exhale out under small thermal-induced pressure difference. Thus, the internal pressure variation ΔP in is regulated by the Clapeyron equation P in V=T ¼ C 1 through thermal expansion and contraction of air inside the microbubble, where C 1 is constant. Since the silica thermal expansion coefficient is very small (%0.5 ppm°C À1 ), [45] we assume that the volume V is constant and obtain 1=T ⋅ dP in =dT À P in =T 2 ⋅ dT=dT ¼ 0 by differentiating operation of Clapeyron equation, which result in ΔP in ¼ P in =T ⋅ dT. This effect is equivalent to the simultaneous change of axial and circumferential stress [44] The microbubble radius change ΔR is affected by the silica thermal expansion and axial force ΔF, which can be expressed with ΔR ¼ R ⋅ α b , where α b is the deformation factor that obtained by a comparison measurement experiment.
where α s is the temperature deformation coefficient of singleended fix microbubble, λ s=b is the optical WGM resonancewavelength of single/both-ended fix microbubble, and Δλ s=b is the spectrum change with temperature. Figure 1. Optomechanical microbubble sensor for temperature sensing. a) Schematic diagram of four effects affecting temperature sensing. Changing the surrounding temperature (T ) causes 1) internal pressure variation ΔP in , 2) Young's modulus variation ΔE, 3) microbubble radius variation ΔR, and 4) axial force variation ΔF. All four effects will change the mechanical-mode frequency. Inset: sectional and top views of the sensor. A cu/si is cross-sectional area of copper sheet/microbubble. b) Schematic diagram of the radiation-pressure-driven opto-mechanical oscillation. c) Illustration of the radiation pressure drives optomechanical oscillators in optical frequency domain. d) Temperature-sensing path of opticalmechanical oscillation mechanism.

Experimental Setup
The wall-thickness-controlled microbubbles used in our experiment are fabricated from commercial fused silica capillary preforms that are tapered under heating with oxyhydrogen flame (see details in Experimental Section). The microbubble used in double clamped sensor has an outer radius of 103 μm and a wall thickness of 1.70 μm (Figure 2a inset), where acoustic and optical modes are simultaneously confined. The temperature sensor is placed on a heating station with high precision temperature controller. Figure 2a illustrates the experimental setup for temperature sensing of these hollow-shell oscillators. This system is used to verify the sensing principle, and the subsequent application only needs C-band laser pumping and demodulation by the data acquisition card. The light from a tunable laser (Keysight 81607 A, linewidth: <10 kHz, tunable range: 1500-1600 nm) is coupled into the microbubble by a tapered optical fiber. Light is evanescently coupled into the WGM mode of microbubble resonator. [46] The input optical power ranges from 1 to 5 mW. The tapered fiber was fabricated by heating a single-mode optical fiber using hydrogen torch while elongating the fiber. The typical diameter of waist is 1-2 μm. The tapered fiber is placed in contact with the microbubble to avoid couple distance affecting. Although the contact may increase the mechanical modes damping and reduce the mechanical Q factors, this effect is very small. [47] The circulation of light energy to and from microbubble couples back to the tapered fiber.
Then, it travels to an optical powermeter module (Keysight 81636B) and a photodetector (Thorlabs PDB450C) through a 1 Â 2 optical switch. The photodetector is connected to an electrical spectrum analyzer (Agilent N9010A) to observe the oscillation frequency spectrum directly. It is worth to point that both the functions of an optical powermeter module and electrical spectrum analyzer with photodetector can be realized by connecting a single data acquisition card to a photodetector, such as National Instruments (NI) product number: PCIÀ5154 that has maximum sample rate of 2 Gs s À1 for signal with frequency measure range upper limit 200 MHz. Since we only need to track the movement of lowest optomechanical oscillation frequency (%10 MHz) during temperature-sensing experiment, a lower sample rate data acquisition card (such as %100 MSPs) can be used and the frequency can be calculated with Fourier transfer of time-domain data as future field sensing solution.
We measure the optical WGM transmission spectrum of microbubble with temperature being controlled at 26°C and determine the optical wavelength of pump light through the analysis of the deepest dip (red point in Figure 2b). Figure 2c shows excited breathing mode when the power of pump light is set as 5 mW, as showed in pervious works. [38,48] The fundamental frequency component is the mechanical eigenfrequency of breathing mode. The high-order harmonic frequency components are related to phase modulation of breathing mode on multiple circulating pump light in microbubble. www.advancedsciencenews.com www.adpr-journal.com

Temperature Measurement
We first investigate the effect of internal pressure variation ΔP in , radius variation ΔR, and Young's modulus variation ΔE si by fixing only single end of microbubble on copper sheet base, which avoid the effect of ΔF. A fabricated microbubble with 100.3 μm radius and 2 μm wall thickness is used to construct the temperature sensor #1 (L ¼ 6 cm, distance from UV glue to the other end of microcapillary). The mechanical-mode frequency is 10.535 MHz at 26°C. The electrical spectrum analyzer measurement range is set as 10.525-10.550 MHz, and the resolution bandwidth (RBW) is 50 Hz. Temperature rises from 26 to 26.7°C with step 0.1°C, and each measurement is carried out after 1 min temperature stabilizing. Figure 3a shows an obviously frequency shift of mechanical mode, and Figure 3b shows a good linear response with a temperature sensitivity of 2700 Hz°C À1 . Figure 3c shows measurement results in heating and cooling operations, which shows good measurement consistency. Using silica material E si ¼ 73.3 GPa, ρ ¼ 2200 kg m À3 , μ ¼ 0.17, the theoretical frequency f 0 will be 10.5461 MHz when n is 14 according to Equation (1), which is close to measurement mechanical-mode frequency 10.535 MHz. Due to the thermal expansion coefficient of silica is extremely small, the ΔR effect is negligible in single-ended fixing configuration, comparing to that of both-ended fixing configuration. Let the ambient temperature be 26°C, and the internal pressure 1 bar in Equation (3), , which is close to measured sensitivity. The actual pressure change may be smaller than theoretically calculated ΔP in , which will lead to the difference between theoretical and experimental sensitivity. We then investigate the designed sensor temperature response with enhanced configuration, both ends of the microbubble are fixed on a copper sheet base. A fabricated microbubble with 103 μm radius and 1.70 μm wall thickness is used to construct the temperature sensor #2 (L ¼ 2.5 cm, distance from two UV glues). Because the radius variation effect ΔR is obtained by comparison measurement experiment, we measured the single-ended and both-ended microbubble WGM optical spectral shifts with temperature. Single-ended result is measured during sensor fabrication, both-ended result is measured after the sensor normally fabricated. As shown in Figure 4, the WGM optical spectral shift is Δλ s =ΔT ¼ 9.55 pm°C À1 and Δλ b =ΔT ¼ À15 pm°C À1 . The experimental results display that the single-ended resonance wavelength has a redshift when temperature rises, which is comparable to those reported in the literature for 6 ppm°C À1 . [18] But for the both-ended, WGM resonance peak blueshifts as temperature increases. The radius variation effect ΔR can be obtained as À15 ppm°C À1 .
We use the same method to excite the optomechanical oscillation and use its frequency spectrum for temperature-sensing experiments. A representative mechanical-mode frequency is 9.774 MHz at 26°C, as shown in Figure 5a. The electrical spectrum analyzer measurement range is set as 9.75-9.80 MHz, and the RBW is 50 Hz. Temperature rises from 26 to 26.6°C with step 0.1°C, and each measurement is carried out after 1 min temperature stabilizing. The black points in Figure 5a are original measurement data, the red lines are fitted data. It can be found that the mechanical-mode frequency increases significantly (marked by blue line). Figure 5b shows a good linear response between the mechanical oscillation frequency shift and the temperature change. The measurement temperature sensitivity is 11 300 Hz°C À1 . This is four times higher than the previous sensitivity 2700 Hz°C À1 and demonstrates that the sensitivity can be improved by ΔF effect. Using the same silica material parameters, the theoretical frequency f 0 will be 9.7765 MHz when n is 15 according to Equation (1), which is close to measurement mechanical-mode frequency 9.774 MHz. ΔP in ¼ 3846.15 Pa°C À1 is the same as sensor #1. Using the cross-sectional area of copper sheet A cu ¼ 8.2 Â 1 mm, the axial force effect is ΔF ¼ 0.0013 N. In the case of four effects, the theoretical temperature sensitivity S T2 ¼ 10 494 Hz°C À1 . The difference between theoretical and experimental sensitivity may be caused by the copper thermal expansion coefficient. The theoretical calculation uses the thermal expansion coefficient of pure copper. However, the experimental copper sheet contains alloy composition, so the actual thermal expansion coefficient is large. If we use the value 18.6 ppm°C À1 near brass coefficient, the theoretical temperature sensitivity is 11 335 Hz°C À1 . Under an optomechanical oscillation mode with full width at half maximum (FWHM) of 114 Hz (see Figure 5c), we can easily locate the peak of optomechanical oscillation in frequency spectrum to Δυ min ¼ FWHM=100, [19,49] corresponding to the spectral resolution of our system, which is about 1.1 Hz. Estimated the resolution ΔT rso of temperature sensor #2 to be 1 Â 10 À4°C accordingly (ΔT rso ¼ Δυ min =S t , S t is temperature sensitivity of sensor). Through the combination of optical WGM spectral direct demodulation and optomechanical oscillation frequency demodulation in same microbubble, we can extend the sensing temperature range while keeping high-temperature resolution. The two-stage method of optical   www.advancedsciencenews.com www.adpr-journal.com scanning first and then precise excitation of optomechanical oscillation can integrate temperature sensors with large dynamic range and ultrahigh resolution. Table 1 summarizes resolution comparison of WGM-based temperature sensors. Compared to similar devices simply observing the WGM optical spectrum, our sensor provides the best temperature resolution level. Only the polydimethylsiloxane (PDMS)-microcavities [19,20] show the same order temperature resolution with our devices. However, PDMS materials have high optical loss regions in the 1100-2500 nm spectral. [50] The increase of optical loss will reduce the Q value, thereby reducing the resolution. Especially in the most commonly used C-band, the absorbance of a 10 mm sample at 1536 nm is 0.33, which is 0.15 at 1463 nm. [51] Most importantly, due to the hysteresis, the PDMS sensor will experience different heating and cooling paths. [26] In addition, the thermal conductivity (0.18 W m À1 * K À1 ) of the PDMS sensor is lower than that of glass (1.05 W m À1 * K À1 ), which may slow response for applications with rapid temperature variation.

Conclusion
In conclusion, we have experimentally demonstrated and characterized the first, to our knowledge, radiation-pressure-driven optomechanical oscillation microresonator used in temperature sensing. The addition of optomechanical oscillation make these sensors much higher resolution to temperature changes than traditional WGM spectrum shift method, and the microbubble does not require common methods such as liquid injection, coated, etc. Observed thermal sensitivity is 11 300 Hz°C À1 and ultrahigh-temperature resolution is 1 Â 10 À4°C . Combined with optical WGM, we provide a new approach to large dynamic range, accurate temperature sensing. Finally, we also present a unique and exciting opportunity to continuous tuning of the discrete optomechanical oscillation frequencies.

Experimental Section
Sensor Fabrication: Wall-thickness-controlled microbubbles were fabricated by stretching and pressurizing microcapillaries in a hightemperature molten state. [53] The fabricating system consisted of three parts, the fused tapering devices, the pressure control devices, and the hydrogen generator. The fabricating process was divided into two steps. First, the original microcapillary was drawn when heated by oxyhydrogen flame with a centimeter scale diameter. Second, we used oxyhydrogen flame with a millimeter scale diameter to heat the waist and a syringe was used to precisely inject air into the capillary, which made the waist swell and develop into a microbubble. During the swelling process, the diameter and the wall thickness could be controlled by adjusting the volume of injected air.
Characterization of the Radius Effect: When temperature rises, the WGM resonance wavelength shift of microbubble is described as [18,54] Δλ where κ ¼ Δn=ΔT is the thermo-optical coefficient and α ¼ ð1=RÞ⋅ ðΔR=ΔTÞ is the deformation coefficient; n and R are the refractive index and the radius of the microbubble, respectively; and λ is the WGM resonance wavelength. For optical WGM spectrum, the difference between single-and double-ended fixations was whether the copper sheet thermal expansion would affect the microbubble deformation, namely the deformation coefficient α. In this case, the single-and both-ended wavelength shift equations are where α s ¼ α si ¼ 0.5 ppm°C À1 . When we measure the Δλ and λ, the bothended deformation coefficient is the radius variation effect (ppm°C À1 ) into the cavity is therefore Characterization of the Axial Force Effect: The axial force ΔF had the opposite effect on copper sheet and cylindrical shell model. However, the deformations of the two are equal and can be written as where ΔL cu=si is the deformation of copper sheet/microtube, ΔL cu ¼ ΔL si ; α cu=si is the thermal expansion coefficient; and A cu/si is the cross-sectional 1.75 Â 10 À3 Ref. [25] www.advancedsciencenews.com www.adpr-journal.com area of copper sheet/microtube (shown in Figure 1a). E cu ¼ 120 GPa,α cu ¼ 16.5 ppm°C À1 . [55] The axial force variation can be written as A si E si (9)